COMPACTNESS IN INTUITIONISTIC FUZZY
TOPOLOGICAL SPACES
A. A. RAMADAN, S. E. ABBAS, AND A. A. ABD EL-LATIF
Received 4 April 2004 and in revised form 17 September 2004
We introduce fuzzy almost continuous mapping, fuzzy weakly continuous mapping,
fuzzy compactness, fuzzy almost compactness, and fuzzy near compactness in intuitionistic fuzzy topological space in view of the definition of Šostak, and study some of their
properties. Also, we investigate the behavior of fuzzy compactness under several types of
fuzzy continuous mappings.
1. Introduction and preliminaries
The concept of a fuzzy set was introduced by Zadeh [13], and later Chang [3] defined
fuzzy topological spaces. These spaces and their generalizations are later studied by several
authors, one of which, developed by Šostak [11, 12], used the idea of degree of openness.
This type of generalization of fuzzy topological spaces was later rephrased by Chattopadhyay et al. [4], and by Ramadan [10].
In 1983, Atanassov introduced the concept of “Intuitionistic fuzzy set” [1, 2]. Using
this type of generalized fuzzy set, Çoker [5, 8] defined “Intuitionistic fuzzy topological
spaces.”
In 1996, Çoker and Demirci [7] introduced the basic definitions and properties of
intuitionistic fuzzy topological spaces in Šostak’s sense, which is a generalized form of
“fuzzy topological spaces” developed by Šostak [11, 12].
In this paper, we introduce the follwing concepts: fuzzy almost continuous mapping,
fuzzy weakly continuous mapping, fuzzy compactness, fuzzy almost compactness, and
fuzzy near compactness in intuitionistic fuzzy topological spaces in view of the definition
of Šostak.
Definition 1.1 [1]. Let X be a nonempty fixed set and I the closed unit interval [0,1]. An
intuitionistic fuzzy set (IFS) A is an object having the form
A=
x,µA (x),νA (x) : x ∈ X ,
(1.1)
where the mappings µA : X → I and νA : X → I denote the degree of membership (namely,
µA (x)) and the degree of nonmembership (namely, νA (x)) of each element x ∈ X to
Copyright © 2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:1 (2005) 19–32
DOI: 10.1155/IJMMS.2005.19
20
Compactness in intuitionistic fuzzy topological spaces
the set A, respectively, and 0 ≤ µA (x) + νA (x) ≤ 1 for each x ∈ X. The complement of
the IFS A, is A = {x,νA (x),µA (x) : x ∈ X }. Obviously, every fuzzy set A on a nonempty
set X is an IFS having the form
A=
x,µA (x),1 − µA (x) : x ∈ X .
(1.2)
For a given nonempty set X, denote the family of all IFSs in X by the symbol ζ X .
Definition 1.2 [6]. Let X be a nonempty set and x ∈ X a fixed element in X. If r ∈ I0 ,
s ∈ I1 are fixed real numbers such that r + s ≤ 1, then the IFS xr,s = y,xr ,1 − x1−s is
called an intuitionistic fuzzy point (IFP) in X, where r denotes the degree of membership
of xr,s , s the degree of nonmembership of xr,s , and x ∈ X the support of xr,s . The IFP xr,s
is contained in the IFS A (xr,s ∈ A) if and only if r < µA (x), s > γA (x).
Definition 1.3 [6]. (i) An IFP xr,s in X is said to be quasicoincident with the IFS A, denoted
by xr,s qA, if and only if r > γA (x) or s < µA (x). xr,s qA if and only if xr,s ∈ A.
(ii) The IFSs A and B are said to be quasicoincident, denoted by AqB if and only if
there exists an element x ∈ X such that µA (x) > γB (x) or γA (x) < µB (x). If A is not quasicoincident with A, denote Aq̃B. Aq̃B if and only if A ⊆ B.
Definition 1.4 [8]. Let a and b be two real numbers in [0,1] satisfying the inequality
a + b ≤ 1. Then the pair a,b is called an intuitionistic fuzzy pair.
Let a1 ,b1 , a2 ,b2 be two intuitionistic fuzzy pairs. Then define
(i) a1 ,b1 ≤ a2 ,b2 if and only if a1 ≤ a2 and b1 ≥ b2 ;
(ii) a1 ,b1 = a2 ,b2 if and only if a1 = a2 and b1 = b2 ;
(iii) if {ai ,bi : i ∈ J } is a family of intuitionistic fuzzy pairs, then ∨ai ,bi = ∨ai , ∧bi and ∧ai ,bi = ∧ai , ∨bi ;
(iv) the complement of an intuitionistic fuzzy pair a,b is the intuitionistic fuzzy pair
defined by a,b = b,a;
(v) 1∼ = 1,0 and 0∼ = 0,1.
Definition 1.5 [5]. An intuitionistic fuzzy topology (IFT) in Chang’s sense on a nonempty
set X is a family τ of IFSs in X satisfying the following axioms:
(T1 ) 0∼ ,1∼ ∈ τ, where 0∼ = {x,0,1 : x ∈ X } and 1∼ = {x,1,0 : x ∈ X };
(T2 ) G1 ∩ G2 ∈ τ for any G1 ,G2 ∈ τ;
(T3 ) ∪Gi ∈ τ for any arbitrary family {Gi : i ∈ J } ⊆ τ.
In this case, the pair (X,τ) is called Chang intuitionistic fuzzy topological space and each
IFS in τ is known as intuitionistic fuzzy open set in X.
Definition 1.6 [8]. An IFS ξ on the set ζ X is called an intuitionistic fuzzy family (IFF)
on X. In symbols, denote such an IFF in form ξ = µξ ,νξ .
Let ξ be an IFF on X. Then the complemented IFF of ξ on X is defined by ξ ∗ =
µξ ∗ ,νξ ∗ , where µξ ∗ (A) = µξ (A) and νξ ∗ (A) = νξ (A), for each A ∈ ζ X . If τ is an IFF on
X, then for any A ∈ ζ X , construct the intuitionistic fuzzy pair µτ (A),ντ (A) and use the
symbol τ(A) = µτ (A),ντ (A).
A. A. Ramadan et al.
21
Definition 1.7 [7]. An IFT in Šostak’s sense on a nonempty set X is an IFF τ on X satisfying
the following axioms:
(T1 ) τ(0∼ ) = τ(1∼ ) = 1∼ ;
(T2 ) τ(A ∩ B) ≥ τ(A) ∧ τ(B) for any A,B ∈ ζ X ;
(T3 ) τ(∪Ai ) ≥ ∧τ(Ai ) for any {Ai : i ∈ J } ⊆ ζ X .
In this case, the pair (X,τ) is called an intuitionistic fuzzy topological space in Šostak’s
sense (IFTS). For any A ∈ ζ X , the number µτ (A) is called the openness degree of A, while
ντ (A) is called the nonopenness degree of A.
Example 1.8. Let X = {a,b}. Define a mapping τ : ζ X → I × I

1∼
τ(A) = 
min µA (a),µA (b) ,max νA (a),νA (b)
if A ∈ {0∼ ,1∼ },
otherwise.
(1.3)
Then, τ is an IFT in the sense of Šostak and neither a Chang fuzzy topology nor a Chang
IFT.
Definition 1.9 [7]. Let (X,τ) be an IFTS on X. Then the IFF τ ∗ is defined by τ ∗ (A) =
τ(A). The number µτ ∗ (A) = µτ (A) is called the closedness degree of A, while ντ ∗ (A) =
ντ (A) is called the nonclosedness degree of A.
Theorem 1.10 [7]. The IFF τ ∗ on X satisfies the following properties:
(C1 ) τ ∗ (0∼ ) = τ ∗ (1∼ ) = 1∼ ;
(C2 ) τ ∗ (A ∪ B) ≥ τ ∗ (A) ∧ τ ∗ (B) for any A,B ∈ ζ X ;
(C3 ) τ ∗ (∩Ai ) ≥ ∧τ ∗ (Ai ) for any {Ai : i ∈ J } ⊆ ζ X .
Definition 1.11 [7]. Let (X,τ) be an IFTS and A be an IFS in X. Then the fuzzy closure
and fuzzy interior of A are defined by
clα,β (A) = ∩ K ∈ ζ X : A ⊆ K, τ ∗ (K) ≥ α,β ,
intα,β (A) = ∪ G ∈ ζ X : G ⊆ A, τ(G) ≥ α,β ,
where α ∈ I0 = (0,1], β ∈ I1 = [0,1) with α + β ≤ 1.
Theorem 1.12 [7]. The closure and interior operator satisfy the following properties:
(i) A ⊆ clα,β (A);
(ii) intα,β (A) ⊆ A;
(iii) A ⊆ B and α,β ≤ r,s implies clα,β (A) ⊆ clr,s (B);
(iv) A ⊆ B and α,β ≤ r,s implies intα,β (A) ⊆ intr,s (B);
(v) clα,β (clα,β (A)) = clα,β (A);
(vi) intα,β (intα,β (A)) = intα,β (A);
(vii) clα,β (A ∪ B) = clα,β (A) ∪ clα,β (B);
(viii) intα,β (A ∩ B) = intα,β (A) ∩ intα,β (B);
(1.4)
22
Compactness in intuitionistic fuzzy topological spaces
(ix) clα,β (A) = intα,β (A);
(x) intα,β (A) = clα,β (A).
Definition 1.13 [7]. Let (X,τ1 ) and (Y ,τ2 ) be two IFTSs and f : X → Y be a mapping.
Then f is said to be
(i) intuitionistic fuzzy continuous if and only if τ1 ( f −1 (B)) ≥ τ2 (B), for each B ∈ ζ Y ;
(ii) intuitionistic fuzzy open if and only if τ2 ( f (A)) ≥ τ1 (A), for each A ∈ ζ X .
2. Intuitionistic fuzzy almost continuous and intuitionistic fuzzy
weakly continuous mapping
Definition 2.1. Let A be an IFS in an IFTS (X,τ). For α ∈ I0 , β ∈ I1 with α + β ≤ 1, A is
called
(i) (α,β)-intuitionistic fuzzy regular open ((α,β)-IFRO) set of X if intα,β (clα,β A) = A;
(ii) (α,β)-intuitionistic fuzzy regular closed ((α,β)-IFRC) set of X if clα,β (intα,β A) =
A.
Theorem 2.2. Let A be an IFS in an IFTS (X,τ). Then, for α ∈ I0 , β ∈ I1 with α + β ≤ 1.
(i) If A is (α,β)-IFRO(resp., (α,β)-IFRC), set then τ(A) ≥ α,β (resp., τ ∗ (A) ≥ α,β).
(ii) A is (α,β)-IFRO set if and only if A is (α,β)-IFRC set.
Proof. We will prove (ii) only:
A is (α,β)-IFRO ⇐⇒ intα,β clα,β A = A
⇐⇒ clα,β intα,β A = A
⇐⇒ A is (α,β)-IFRC.
(2.1)
Theorem 2.3. Let (X,τ) be an IFTS. Then,
(i) the union of two (α,β)-IFRC sets is (α,β)-IFRC set,
(ii) the intersection of two (α,β)-IFRO sets is (α,β)-IFRO set.
Proof. (i) Let A, B be any two (α,β)-IFRC sets. By Theorem 2.2, we have τ ∗ (A) ≥ α,β,
τ ∗ (B) ≥ α,β then, τ ∗ (A ∪ B) ≥ τ ∗ (A) ∧ τ ∗ (B) ≥ α,β, but intα,β (A ∪ B) ⊆ A ∪ B, this
implies that clα,β (intα,β (A ∪ B)) ⊆ clα,β (A ∪ B) = A ∪ B. Now, A = clα,β (intα,β (A)) ⊆
clα,β (intα,β (A ∪ B)) and B = clα,β (intα,β (B)) ⊆ clα,β (intα,β (A ∪ B)). Then, A ∪ B ⊆
clα,β (intα,β (A ∪ B)). So, clα,β (intα,β (A ∪ B)) = A ∪ B. Hence, A ∪ B is (α,β)-IFRC set.
(ii) It can be proved by the same manner.
Theorem 2.4. Let (X,τ) be an IFTS. Then,
(i) if A ∈ ζ X such that τ ∗ (A) ≥ α,β, then intα,β (A) is (α,β)-IFRO set,
(ii) if B ∈ ζ X such that τ(B) ≥ α,β, then clα,β (B) is (α,β)-IFRC set.
Proof. (i) Let A ∈ ζ X such that τ ∗ (A) ≥ α,β. Clearly,
intα,β (A) ⊆ intα,β clα,β (A) ;
(2.2)
A. A. Ramadan et al.
23
this implies that
intα,β (A) ⊆ intα,β clα,β intα,β (A) .
(2.3)
Now, since τ ∗ (A) ≥ α,β, then clα,β (intα,β (A)) ⊆ A; this implies that
intα,β clα,β intα,β (A)
⊆ intα,β (A).
(2.4)
Thus, intα,β (clα,β (intα,β (A))) = intα,β (A). Hence, intα,β (A) is (α,β)-IFRO set.
(ii) It can be proved by the same manner.
Definition 2.5. A mapping f : (X,τ1 ) → (Y ,τ2 ) from an IFTS (X,τ1 ) to another IFTS
(Y ,τ2 ) is called
(i) intuitionistic fuzzy strong continuous if and only if τ1 ( f −1 (A)) = τ2 (A), for each
A ∈ ζY ,
(ii) (α,β)-intuitionistic fuzzy almost continuous if and only if τ1 ( f −1 (A)) ≥ α,β, for
each (α,β)-IFRO set A of Y ,
(iii) (α,β)-intuitionistic fuzzy weakly continuous if and only if τ2 (A) ≥ α,β implies
τ1 ( f −1 (A)) ≥ α,β, for each A ∈ ζ Y .
Remark 2.6. From the above definition, it is clear that the following implications are true
for α ∈ I0 , β ∈ I1 with α + β ≤ 1:
(α,β)-intuitionistic fuzzy almost continuous mapping
intuitionistic fuzzy strong continuous
intuitionistic fuzzy continuous mapping
(α,β)-intuitionistic fuzzy weakly continuous mapping
(2.5)
But, the reciprocal implications are not true in general, as shown by the following examples.
Example 2.7. Let X = {a,b,c} and G1 , G2 be IFSs in X defined as follows:
G1 = a,0.4,0.1, b,0.6,0.2, c,0.5,0.3 ,
G2 = a,0.4,0.4, b,0.4,0.4, c,0.4,0.4 .
(2.6)
24
Compactness in intuitionistic fuzzy topological spaces
We define an IFTs τ1 ,τ2 : ζ X → I × I as follows:


1∼




0.5,0.2
τ1 (A) = 
0.5,0.3



 ∼
if A ∈ {0∼ ,1∼ },
if A = G1 ,
if A = G2 ,
otherwise,

∼


1

τ2 (A) = 0.6,0.2


0∼
if A ∈ {0∼ ,1∼ },
if A = G2 ,
otherwise.
0
(2.7)
Let α = 0.4, β = 0.5. Then, the identity mapping idX : (X,τ1 ) → (X,τ2 ) is (α,β)-intuitionistic fuzzy almost continuous, but not intuitionistic fuzzy continuous.
Example 2.8. Let X = {a,b}, Y = {1,2}. Let G1 be an IFS of X and G2 be an IFS of Y ,
defined as follows:
G1 = a,0.4,0.4, b,0.4,0.4 ,
G2 = 1,0.4,0.4, 2,0.5,0.4 .
(2.8)
We define an IFTs τ1 : ζ X → I × I and τ2 : ζ Y → I × I as follows:

∼



1
τ1 (A) = 0.7,0.1


0∼
if A ∈ {0∼ ,1∼ },
if A = G1 ,
otherwise,

∼



1
τ2 (A) = 0.8,0.1


0∼
if A ∈ {0∼ ,1∼ },
if A = G2 ,
otherwise.
(2.9)
Consider the mapping f : (X,τ1 ) → (Y ,τ2 ) defined by
f (a) = 1,
f (b) = 1.
(2.10)
Let α = 0.6, β = 0.3. Then, f is (α,β)-intuitionistic fuzzy weakly continuous, but not
intuitionistic fuzzy continuous.
Example 2.9. In the above example, if

∼



1
τ1 (A) = 0.8,0.2


0∼
if A ∈ {0∼ ,1∼ },
if A = G1 ,
otherwise,
(2.11)
then f is intuitionistic fuzzy continuous, but not intuitionistic fuzzy strong continuous.
A. A. Ramadan et al.
25
Theorem 2.10. Let f : (X,τ1 ) → (Y ,τ2 ) be a mapping from an IFTS (X,τ1 ) to another IFTS
(Y ,τ2 ). Then, the following statements are equivalent:
(i) f is (α,β)-intuitionistic fuzzy almost continuous;
(ii) τ1∗ ( f −1 (B)) ≥ α,β, for each (α,β)-IFRC set B of Y ;
(iii) f −1 (B) ⊆ intα,β f −1 (intα,β (clα,β (B))), for each B ∈ ζ Y such that τ2 (B) ≥ α,β;
(iv) clα,β f −1 (clα,β (intα,β (B))) ⊆ f −1 (B), for each B ∈ ζ Y such that τ2 (B) ≥ α,β, where
α ∈ I0 , β ∈ I1 with α + β ≤ 1.
Proof. (i)⇒(ii). Let B be (α,β)-IFRC set of Y . Then, by Theorem 2.2, B is (α,β)-IFRO set.
By, (i), we have τ1 ( f −1 (B)) = τ1 ( f −1 (B)) = τ1∗ ( f −1 (B)) ≥ α,β.
(ii)⇒(i). It is analogous to the proof of (ii)⇒(i).
(i)⇒(iii). Since τ2 (B) ≥ α,β, then B = intα,β (B) ⊆ intα,β (clα,β (B)) and hence, f −1 (B) ⊆
f −1 (intα,β (clα,β (B))) since, τ2∗ (clα,β (B)) ≥ α,β, then by Theorem 2.4, intα,β (clα,β (B)) is
(α,β)-IFRO set. So, τ1 ( f −1 (intα,β (clα,β (B)))) ≥ α,β. Then, f −1 (B) ⊆ f −1 (intα,β (clα,β (B)))
= intα,β ( f −1 (intα,β (clα,β (B)))).
(iii)⇒(i). Let B be (α,β)-IFRO set of Y . Then, we have
f −1 (B) ⊆ intα,β f −1 intα,β clα,β (B)
= intα,β f −1 (B) ;
(2.12)
this implies that f −1 (B) = intα,β ( f −1 (B)), then
τ1 f −1 (B) = τ1 intα,β f −1 (B)
≥ α,β.
Hence, f is (α,β)-intuitionistic fuzzy almost continuous.
(ii)⇔(iv). Can similarly be proved.
(2.13)
Theorem 2.11. Let f : (X,τ1 ) → (Y ,τ2 ) be a mapping from an IFTS (X,τ1 ) to another IFTS
(Y ,τ2 ). Then, the following are equivalent:
(i) f is (α,β)-intuitionistic fuzzy weakly continuous;
(ii) f (clα,β (A)) ⊆ clα,β ( f (A)) for each A ∈ ζ X .
Proof. (i)⇒(ii). Let A ∈ ζ X . Then,
f −1 clα,β f (A)
= f −1 ∩ k ∈ ζ Y : τ2∗ (k) ≥ α,β, k ⊇ f (A)
= f −1 ∩ k ∈ ζ Y : τ2 k ≥ α,β, k ⊇ f (A)
⊇ f −1 ∩ k ∈ ζ Y : τ1∗ f −1 (k) = τ1 f −1 (k) ≥ α,β, k ⊇ f (A)
⊇ ∩ f −1 (k) : k ∈ ζ Y : τ1∗ f −1 (k) ≥ α,β, f −1 (k) ⊇ A
⊇ ∩ G ∈ ζ X : τ1∗ (G) ≥ α,β, G ⊇ A = clα,β (A).
(2.14)
Then, f (clα,β (A)) ⊆ f ( f −1 (clα,β ( f (A)))) ⊆ clα,β ( f (A)).
(ii)⇒(i). Let B ∈ ζ Y such that τ2 (B) ≥ α,β. Then, τ2∗ (B) = τ2 (B) ≥ α,β. So, we have
clα,β (B) = B. Further, since f (clα,β ( f −1 (B))) ⊆ clα,β ( f ( f −1 (B))) ⊆ clα,β (B) = B, we have
26
Compactness in intuitionistic fuzzy topological spaces
clα,β ( f −1 (B)) ⊆ f −1 (B). Then, clα,β ( f −1 (B)) = f −1 (B). This implies that τ1∗ ( f −1 (B)) ≥
α,β, therefore, τ1∗ ( f −1 (B)) = τ1 ( f −1 (B)) ≥ α,β. Hence, f is (α,β)-intuitionistic fuzzy
weakly continuous.
Theorem 2.12. Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect
to the IFTs τ1 and τ2 respectively. Then for every IFS A in X,
f clα,β (A) ⊆ clα,β f (A) ,
(2.15)
where α ∈ I0 , β ∈ I1 with α + β ≤ 1.
Proof. Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect to τ1
and τ2 , and let A ∈ ζ X . Then,
f −1 clα,β f (A)
= f −1 ∩ K ∈ ζ Y , τ2∗ (K) ≥ α,β, f (A) ⊆ K
= ∩ f −1 (K) : K ∈ ζ Y , τ2∗ (K) ≥ α,β, A ⊆ f −1 (K)
⊇ ∩ f −1 (K) : K ∈ ζ Y , τ1∗ f −1 (K) ≥ α,β, A ⊆ f −1 (K)
⊇ ∩ G ∈ ζ X : τ1∗ (G) ≥ α,β, A ⊆ G = clα,β (A).
This implies that f (clα,β (A)) ⊆ clα,β ( f (A)).
(2.16)
Theorem 2.13. Let f : X → Y be an intuitionistic fuzzy continuous mapping with respect
to the IFTs τ1 and τ2 , respectively. Then, for every IFS A in Y ,
clα,β f −1 (A) ⊆ f −1 clα,β (A) ,
(2.17)
where, α ∈ I0 , β ∈ I1 with α + β ≤ 1.
Proof. Let A ∈ ζ Y . We get from Theorem 2.12
clα,β f −1 (A) ⊆ f −1 f clα,β f −1 (A)
⊆ f −1 clα,β (A) .
Hence, clα,β ( f −1 (A)) ⊆ f −1 (clα,β (A)), for every A ∈ ζ Y .
(2.18)
3. Various cases of compactness in intuitionistic fuzzy topological spaces
Definition 3.1. An IFTS (X,τ) is called (α,β)-intuitionistic fuzzy compact (resp., (α,β)intuitionistic fuzzy nearly compact and (α,β)-intuitionistic fuzzy almost compact) if and
only if for every family {Gi : i ∈ J } in {G : G ∈ ζ X , τ(G) > α,β} such that ∪i∈J Gi = 1∼ ,
where α ∈ I0 , β ∈ I1 with α + β ≤ 1, there exists a finite subset J0 of J such that ∪i∈J0 Gi = 1∼
(resp., ∪i∈J0 intα,β (clα,β (Gi )) = 1∼ and ∪i∈J0 clα,β (Gi ) = 1∼ ).
Definition 3.2. Let (X,τ) be an IFTS and A an IFS in X. A is said to be (α,β)-intuitionistic
fuzzy compact if and only if every family {Gi : i ∈ J } in {G : G ∈ ζ X , τ(G) > α,β} such
that A ⊆ ∪i∈J Gi , there exists a finite subset J0 of J such that A ⊆ ∪i∈J0 Gi , where α ∈ I0 ,
β ∈ I1 with α + β ≤ 1.
A. A. Ramadan et al.
27
Example 3.3. Let X = I and consider the IFSs {Gn : n = 2,3,4, ...} as follows: first we
define IFSs Gn = x,µGn ,νGn and G = x,µG ,νG by


0.8,




µGn (x) = nx,



1,
x = 0,
1
0<x≤ ,
n
1
< x ≤ 1,
n


0.1,




νGn (x) = 1 − nx,



0,

0.8,
µG (x) = 
1,

0.1,
νG (x) = 
0,
x = 0,
1
0<x≤ ,
n
1
< x ≤ 1,
n
(3.1)
x = 0,
otherwise,
x = 0,
otherwise.
Second, we define the IFT τ : ζ X → I × I as follows:


1∼







 1 1
,
τ(A) =  n 2n



0.7,0.2




0∼
if A ∈ {0∼ ,1∼ },
if A = Gn ,
(3.2)
if A = G,
otherwise.
Let α = 0.6, β = 0.2. Then, the IFS C0.85,0.15 = {x,0.85,0.15 : x ∈ X } is (α,β)-intuitionistic fuzzy compact and the IFS C0.75,0.15 = {x,0.75,0.15 : x ∈ X } is not (α,β)-intuitionistic fuzzy compact.
Theorem 3.4. For α ∈ I0 , β ∈ I1 with α + β ≤ 1, (α,β)-intuitionistic fuzzy compactness implies (α,β)-intuitionistic fuzzy nearly compactness which implies (α,β)-intuitionistic fuzzy
almost compactness.
Proof. Let an IFTS (X,τ) be (α,β)-intuitionistic fuzzy compact. Then, for every family
{Gi : i ∈ J } in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1 with α + β ≤ 1 such that
∪i∈J Gi = 1∼ , there exists a finite subset J0 of J such that ∪i∈J0 Gi = 1∼ . Now, since τ(Gi ) >
α,β for each i ∈ J, then Gi = intα,β Gi for each i ∈ J. Also, Gi = intα,β Gi ⊆ intα,β (clα,β Gi )
for each i ∈ J. Then, 1∼ = ∪i∈J0 Gi = ∪i∈J0 intα,β Gi ⊆ ∪i∈J0 intα,β (clα,β Gi ). Thus,
∪i∈J0 intα,β (clα,β Gi ) = 1∼ . Hence, an IFTS (X,τ) is (α,β)-intuitionistic fuzzy nearly com-
pact.
28
Compactness in intuitionistic fuzzy topological spaces
For the second implication, suppose that the IFTS (X,τ) is (α,β)-intuitionistic fuzzy
nearly compact, then for every family {Gi : i ∈ J } in {G : G ∈ ζ X , τ(G) > α,β}, where
α ∈ I0 , β ∈ I1 with α + β ≤ 1, there exists a finite subset J0 of J such that ∪i∈J0 intα,β (clα,β Gi )
= 1∼ since, Gi = intα,β Gi ⊆ intα,β clα,β Gi ⊆ clα,β Gi for each i ∈ J then, 1∼ =
∪i∈J0 intα,β clα,β Gi ⊆ ∪i∈J0 clα,β Gi . Thus, ∪i∈J0 clα,β Gi = 1∼ . Hence, the IFTS (X,τ) is (α,β)intuitionistic fuzzy almost compact.
Remark 3.5. In IFTS in Chang’s sense, the converse of these two implications are not
valid for compactness, nearly compactness, and almost compactness [9], which are special cases of compactness, nearly compactness and almost compactness, respectively in
IFTS, in Šostak’s sense. Thus, the converse implications in Theorem 3.4 are not true in
general.
Definition 3.6. A family {Ki : i ∈ J } in {K : K ∈ ζ X , τ ∗ (K) > α,β}, where α ∈ I0 , β ∈ I1
with α + β ≤ 1 has the finite intersection property (FIP) if and only if for any finite subset
J0 of J, ∩i∈J0 Ki = 0∼ .
Theorem 3.7. An IFTS (X,τ) is (α,β)-intuitionistic fuzzy compact, if and only if every
family in {K : K ∈ ζ X , τ ∗ (K) > α,β}, where, α ∈ I0 , β ∈ I1 with α + β ≤ 1 having the FIP,
has a nonempty intersection.
Proof. Let an IFTS (X,τ) be (α,β)-intuitionistic fuzzy compact, and consider the family
{Ki : i ∈ J } in {K : K ∈ ζ X , τ ∗ (K) > α,β} having the FIP. Now suppose that ∩i∈J Ki = 0∼
then, ∪i∈J Ki = 1∼ , from τ(Ki ) = τ ∗ (Ki ) > α,β and (X,τ) is (α,β)-intuitionistic fuzzy
compact, we have ∪i∈J0 Ki = 1∼ ; this implies that ∩i∈J0 Ki = 0∼ , which is a contradiction.
Conversely, let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 ,
β ∈ I1 with α + β ≤ 1 such that ∪i∈J Gi = 1∼ . If ∪i∈J0 Gi = 1∼ for every finite subset J0 of
J, then ∩i∈J0 Gi = 0∼ and the family {Gi : i ∈ J } has the FIP and hence from the given
condition, we have ∩i∈J Gi = 0∼ so, ∪i∈J Gi = 1∼ , a contradiction.
Definition 3.8. An IFTS (X,τ) is called (α,β)-intuitionistic fuzzy regular if and only if for
each IFS A in X such that τ(A) > α,β, where α ∈ I0 , β ∈ I1 with α + β ≤ 1, can be written
as A = ∪{B : B ∈ ζ X , τ(B) ≥ τ(A), clα,β (B) ⊆ A}.
Theorem 3.9. Let (X,τ) be an IFTS. If (X,τ) is (α,β)-intuitionistic fuzzy almost compact
and (α,β)-intuitionistic fuzzy regular, then it is (α,β)-intuitionistic fuzzy compact.
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1
with α + β ≤ 1 such that ∪i∈J Gi = 1∼ . From the fuzzy regularity of (X,τ), it follows that
Gi = ∪{Bi : Bi ∈ ζ X , τ(Bi ) ≥ τ(Gi ), clα,β (Bi ) ⊆ Gi }. Since ∪i∈J Gi = ∪i∈J Bi = 1∼ , τ(Bi ) ≥
τ(Gi ) > α,β then from almost compactness of (X,τ) there exists a finite subset J0 of J
such that ∪i∈J0 clα,β (Bi ) = 1∼ . But clα,β (Bi ) ⊆ Gi , this implies that ∪i∈J0 Gi ⊇ ∪i∈J0 clα,β (Bi ) =
1∼ that implies ∪i∈J0 Gi = 1∼ . Hence, (X,τ) is (α,β)-intuitionistic fuzzy compact.
Theorem 3.10. Let (X,τ) be an IFTS. If (X,τ) is (α,β)-intuitionistic fuzzy nearly compact
and (α,β)-intuitionistic fuzzy regular, then it is (α,β)-intuitionistic fuzzy compact.
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1 with
α + β ≤ 1 such that ∪i∈J Gi = 1∼ .
A. A. Ramadan et al.
29
From fuzzy regularity of (X,τ) it follows that Gi = ∪{Bi : Bi ∈ ζ X , τ(Bi ) ≥ τ(Gi ),
clα,β (Bi ) ⊆ Gi } then, 1∼ = ∪i∈J Gi = ∪i∈J Bi , τ(Bi ) ≥ τ(Gi ) > α,β. Since (X,τ) is (α,β)intuitionistic fuzzy nearly compact, there exists a finite subset J0 of J such that
∪i∈J0 intα,β (clα,β Gi ) = 1∼ . But, intα,β (clα,β Bi ) ⊆ clα,β Bi ⊆ Gi then, ∪i∈J0 Gi = 1∼ .
Hence, (X,τ) is (α,β)-intuitionistic fuzzy compact.
Theorem 3.11. An IFTS (X,τ) is (α,β)-intuitionistic fuzzy almost compact if and only if
every family {Gi : i ∈ J } in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1 with α + β ≤ 1
having the FIP, ∩i∈J clα,β Gi = 0∼ .
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1
with α + β ≤ 1 having the FIP. Suppose that ∩i∈J clα,β Gi = 0∼ , then we have ∪i∈J clα,β Gi =
∪i∈J intα,β Gi = 1∼ . Since τ(intα,β Gi ) ≥ α,β and X is (α,β)-intuitionistic fuzzy almost
compact, there exists a finite subset J0 of J such that ∪i∈J0 clα,β (intα,β Gi ) = 1∼ this implies that ∪i∈J0 clα,β (intα,β Gi ) = ∪i∈J0 clα,β (clα,β Gi ) = ∪i∈J0 intα,β (clα,β Gi ) = 1∼ . Thus,
∩i∈J0 intα,β clα,β Gi = 0∼ , but from Gi = intα,β Gi ⊆ intα,β clα,β Gi , we see that ∩i∈J0 Gi = 0∼ ,
which is a contradiction with the FIP of the family.
Conversely, let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 ,
β ∈ I1 with α + β ≤ 1 such that ∪i∈J Gi = 1∼ . Suppose that there exists no finite subset
J0 of J such that ∪i∈J0 clα,β Gi = 1∼ . Since τ(clα,β Gi ) = τ ∗ (clα,β Gi ) ≥ α,β, then by our
hypothesis, the family {clα,β Gi : i ∈ J } has the FIP. So, we have
∩i∈J clα,β clα,β Gi = 0∼
=⇒ ∪i∈J clα,β clα,β Gi = 1∼
=⇒ ∪i∈J intα,β clα,β Gi = 1∼ ,
(3.3)
since Gi ⊆ intα,β (clα,β Gi ), for each i ∈ J then, ∪i∈J Gi = 1∼ , which is a contradiction with
∪i∈J Gi = 1∼ .
Lemma 3.12. Let (X,τ) be an IFTS and V ∈ ζ X . Then, xr,s ∈ clα,β V if and only if for each
U ∈ ζ X with τ(U) ≥ α,β and xr,s qU, UqV , where r,α ∈ I0 , s,β ∈ I1 with r + s, α + β ≤ 1.
Proof. Let xr,s ∈ clα,β V and let U be any IFS in X such that τ(U) ≥ α,β and xr,s qU.
Suppose for a contradiction that V q̃U. Then, we have V ⊆ U. Since xr,s qU, then xr,s ∈
U ⊇ V , and since τ ∗ (U) = τ(U) ≥ α,β, then xr,s ∈ clα,β V , which is a contradiction,
then V qU. Conversely, suppose that for any U ∈ ζ X with τ(U) ≥ α,β such that xr,s qU,
we have UqV . Suppose, for a contradiction, that xr,s ∈ clα,β V . Then, there exists B ∈ ζ X
with τ ∗ (B) ≥ α,β, B ⊇ V and xr,s ∈ B. Thus, τ(B) = τ ∗ (B) ≥ α,β and xr,s qB. Then,
from our hypotheses V qB, which implies that V ⊆ B; this is a contradiction. Hence, xr,s ∈
clα,β V .
Lemma 3.13. Let (X,τ) be an IFTS. For α ∈ I0 , β ∈ I1 with α + β ≤ 1, clα,β (intα,β (clα,β A)) =
clα,β A, for each A ∈ ζ X with τ(A) ≥ α,β.
Proof. Let A ∈ ζ X with τ(A) ≥ α,β. A ⊆ clα,β A implies A ⊆ intα,β (clα,β A). Then,
clα,β A ⊆ clα,β intα,β clα,β A .
(3.4)
30
Compactness in intuitionistic fuzzy topological spaces
Let xt,s ∈ clα,β A, where r ∈ I0 , s ∈ I1 with r + s ≤ 1. Then by Lemma 3.12, there exists
U ∈ ζ X with τ(U) ≥ α,β such that xt,s qU and Aq̃U. From Aq̃U, it follows that A ⊆ U,
so using the fact that τ ∗ (U) = τ(U) ≥ α,β, we obtain that clα,β A ⊆ U. Thus clα,β Aq̃U.
This implies that intα,β (clα,β A)q̃U. Since τ(U) ≥ α,β and xt,s qU, then by Lemma 3.12,
we have xt,s ∈ clα,β (intα,β (clα,β A)). Thus
clα,β intα,β clα,β A
⊆ clα,β A.
(3.5)
= clα,β A.
(3.6)
From (3.4) and (3.5) we have
clα,β intα,β clα,β A
Theorem 3.14. In an IFTS (X,τ) the following conditions are equivalent.
(i) (X,τ) is (α,β)-intuitionistic fuzzy almost compact.
(ii) For every family G = {Gi : i ∈ J }, where Gi = {x,µGi ,νGi : i ∈ J } of (α,β)-IFRC sets
such that ∩i∈J Gi = 0∼ , there exists a finite subset J0 of J such that ∩i∈J0 intα,β Gi = 0∼ , where
α ∈ I0 , β ∈ I1 with α + β ≤ 1.
(iii) ∩i∈J clα,β Gi = 0∼ holds for every family {Gi ∈ ζ X : i ∈ J } of (α,β)-IFRO sets having
the FIP, where α ∈ I0 , β ∈ I1 with α + β ≤ 1.
(iv) For every family {Gi ∈ ζ X : i ∈ J } of (α,β)-IFRO sets such that ∪i∈J Gi = 1∼ , there
exists a finite subset J0 of J such that ∪i∈J0 clα,β Gi = 1∼ .
Proof. (i)⇒(ii). Let G = {Gi : i ∈ J } be a family of (α,β)-IFRC sets in X with ∩i∈J Gi = 0∼ .
Then, ∪i∈J Gi = 1∼ . By, Theorem 2.2, Gi is (α,β)-IFRO set then, ∪i∈J intα,β (clα,β Gi ) = 1∼ .
Since τ(intα,β (clα,β Gi )) ≥ α,β and (X,τ) is (α,β)-intuitionistic fuzzy almost compact
then, there exists a finite subset J0 of J such that ∪i∈J0 clα,β (intα,β (clα,β Gi )) = 1∼ , this implies
that ∪i∈J0 clα,β (intα,β (clα,β Gi )) = ∩i∈J0 clα,β (intα,β (clα,β Gi )) = ∩i∈J0 intα,β (clα,β (intα,β Gi )) =
∩i∈J0 intα,β Gi = 0∼ .
(ii)⇒(iii). Let {Gi ∈ ζ X : i ∈ J } be (α,β)-IFRO sets having the FIP and suppose that
∩i∈J clα,β Gi = 0∼ . Since Gi is (α,β)-IFRO set, then by Theorem 2.2, τ(Gi ) ≥ α,β and by
Theorem 2.4, we have {clα,β Gi : i ∈ J } is a family of (α,β)-IFRC sets then, by (ii), there
exists a subset J0 of J such that ∩i∈J0 intα,β (clα,β Gi ) = ∩i∈J0 Gi = 0∼ , which is a contradiction.
(iii)⇒(iv). Let {Gi ∈ ζ X : i ∈ J } be a family of (α,β)-IFRO sets such that ∪i∈J Gi =
1∼ . Suppose that for every finite subset J0 of J, ∪i∈J0 clα,β Gi = 1∼ . Then, by Theorems
2.2 and 2.4, we have {clα,β (Gi ) : i ∈ J } is a family of (α,β)-IFRO sets having the FIP.
Since, intα,β (clα,β (clα,β (Gi ))) = intα,β (intα,β (clα,β (Gi ))) = intα,β Gi = clα,β Gi , hence by (iii)
∩i∈J clα,β (clα,β Gi ) = 0∼ implies ∩i∈J intα,β (clα,β Gi ) = 0∼ which implies ∩i∈J Gi = 0∼
which implies ∪i∈J Gi = 1∼ which is a contradiction with ∪i∈J Gi = 1∼ .
(iv)⇒(i). Let {Gi : i ∈ J } be a family in {G : G ∈ ζ X , τ(G) > α,β}, where α ∈ I0 , β ∈ I1
with α + β ≤ 1 such that ∪i∈J Gi = 1∼
Gi ⊆ clα,β Gi =⇒ Gi ⊆ intα,β clα,β Gi .
(3.7)
A. A. Ramadan et al.
31
Thus, ∪i∈J intα,β (clα,β Gi ) = 1∼ . Since intα,β (clα,β Gi ) is (α,β)-IFRO, then by (iv), there exists a finite subset J0 of J such that ∪i∈J0 clα,β (intα,β (clα,β Gi )) = 1∼ . By using Lemma 3.13,
we have, ∪i∈J0 clα,β Gi = 1∼ . Hence, (X,τ) is (α,β)-intuitionistic fuzzy almost compact.
Theorem 3.15. Let (X,τ1 ), (Y ,τ2 ) be two IFTSs and f : X → Y an intuitionistic fuzzy
continuous mapping. If A is (α,β)-intuitionistic fuzzy compact in (X,τ1 ) then, so is f (A) in
(Y ,τ2 ), where α ∈ I0 , β ∈ I1 with α + β ≤ 1.
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ Y , τ2 (G) > α,β}, where α ∈ I0 , β ∈ I1 with
α + β ≤ 1 such that f (A) ⊆ ∪i∈J Gi . Then, A ⊆ f −1 ( f (A)) ⊆ f −1 (∪i∈J Gi ) = ∪i∈J f −1 (Gi ).
Since f is an intuitionistic fuzzy continuous, then τ1 ( f −1 (Gi )) ≥ τ2 (Gi ) > α,β and since
A is (α,β)-intuitionistic fuzzy compact in (X,τ1 ), there exists a finite subset J0 of J such
that A ⊆ ∪i∈J0 f −1 (Gi ); this implies that f (A) ⊆ f (∪i∈J0 f −1 (Gi )) = ∪i∈J0 f ( f −1 (Gi )) ⊆
∪i∈J0 Gi . Hence, f (A) is (α,β)-intuitionistic fuzzy compact in (Y ,τ2 ).
Theorem 3.16. Let (X,τ1 ), (Y ,τ2 ) be two IFTSs and f : X → Y a surjection intuitionistic
fuzzy continuous. If (X,τ1 ) is (α,β)-intuitionistic fuzzy compact, then so is (Y ,τ2 ), where
α ∈ I0 , β ∈ I1 with α + β ≤ 1.
The proof is similar to that of Theorem 3.15.
Theorem 3.17. Let f : (X,τ1 ) → (Y ,τ2 ) be a surjective intuitionistic fuzzy continuous mapping with respect to τ1 and τ2 , respectively. If (X,τ1 ) is (α,β)-intuitionistic fuzzy almost compact, then so is (Y ,τ2 ), where α ∈ I0 , β ∈ I1 with α + β ≤ 1.
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ Y , τ2 (G) > α,β} such that ∪i∈J Gi = 1∼ ,
where α ∈ I0 , β ∈ I1 with α + β ≤ 1. Since f is intuitionistic fuzzy continuous, we have
τ1 ( f −1 (Gi )) ≥ τ2 (Gi ) > α,β but, ∪i∈J f −1 (Gi ) = f −1 (∪i∈G Gi ) = 1∼ .
Then from the almost compactness of (X,τ1 ), there exists a subset J0 of J such that
∪i∈J0 clα,β ( f −1 (Gi )) = 1∼ , this implies that
f ∪i∈J0 clα,β f −1 Gi
= ∪i∈J0 f clα,β f −1 Gi
= 1∼ .
(3.8)
But from Theorem 2.13, we have f −1 (clα,β (Gi )) ⊇ clα,β ( f −1 (Gi )) and, from the surjectivity of f , we have clα,β (Gi ) = f ( f −1 (clα,β (Gi ))) ⊇ f (clα,β ( f −1 (Gi ))). So, ∪i∈J0 clα,β (Gi ) ⊇
∪i∈J0 f (clα,β ( f −1 (Gi ))) = 1∼ . Then, ∪i∈J0 clα,β (Gi ) = 1∼ . Hence, (Y ,τ2 ) is (α,β)-intuition
istic fuzzy almost compact.
Theorem 3.18. Let (X,τ1 ) and (Y ,τ2 ) be IFTSs and let f : X → Y be (α,β)-intuitionistic
fuzzy weakly continuous surjection mapping. If (X,τ1 ) is (α,β)-intuitionistic fuzzy compact,
then (Y ,τ2 ) is (α,β)-intuitionistic fuzzy almost compact.
Proof. Let {Gi : i ∈ J } be a family in {G : G ∈ ζ Y , τ2 (G) > α,β}, where α ∈ I0 , β ∈ I1
with α + β ≤ 1 such that ∪i∈J Gi = 1∼ . Since f is (α,β)-intuitionistic fuzzy weakly continuous, we have τ1 ( f −1 (Gi )) > α,β then, ∪i∈J ( f −1 (Gi )) = f −1 (∪i∈J Gi ) = f −1 (1∼ ) = 1∼ .
Since (X,τ1 ) is (α,β)-intuitionistic fuzzy compact, there exists a finite subset J0 of J such
that ∪i∈J0 f −1 (Gi ) = 1∼ but, f −1 (clα,β Gi ) ⊇ f −1 (Gi ), this implies that ∪i∈J0 f −1 (clα,β Gi ) ⊇
∪i∈J0 f −1 (Gi ) = 1∼ .
32
Compactness in intuitionistic fuzzy topological spaces
Then, ∪i∈J0 f −1 (clα,β Gi ) = 1∼ . Since f is surjective,
f ∪i∈J0 f −1 clα,β Gi
= ∪i∈J0 f f −1 clα,β Gi = ∪i∈J0 clα,β Gi = 1∼ .
(3.9)
Thus, ∪i∈J0 clα,β Gi = 1∼ . Hence, (Y ,τ2 ) is (α,β)-intuitionistic fuzzy almost compact.
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A. A. Ramadan: Department of Mathematics, Faculty of Sciences, King Saud University, P.O. Box
237, Burieda 81999, Saudi Arabia
E-mail address: aramadan58@yahoo.com
S. E. Abbas: Department of Mathematics, Faculty of Science, South Valley University, Sohag
82524, Egypt
E-mail address: sabbas73@yahoo.com
A. A. Abd El-Latif: Department of Mathematics, Faculty of Science, Cairo University, Beni Suef,
Egypt
E-mail address: ahmeda73@yahoo.com